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Force identification is a classical inverse problem in which the measured data and the mathematical models of mechanical structures are used to determine the applied force. However, the identified force may seriously diverge from the true solution due to the unknown noise contaminating the measured data and the inverse of the ill-posed transfer matrix characterizing the mechanical structure. In this paper, a novel method based on the discrete cosine transform (DCT) in the time domain is proposed for force identification, which overcomes the deficiency of the ill-posedness of the transfer matrix. The unknown force is expanded by a set of cosine basis functions and then the original governing equation is reformulated to find the coefficient of each cosine basis function. Furthermore, a modified generalized cross-validation (GCV) criterion for determining the regularization parameter is developed for the truncated singular value decomposition (TSVD), Chebyshev polynomial and DCT solutions. Numerical simulation reveals that compared with the L-curve criterion, the modified GCV criterion is quite robust in the presence of noise.Finally, a clamped-free shell structure that is excited by an impact hammer is selected as an example to validate the performance of the proposed method. Experimental results demonstrate that compared with the TSVD-based and Chebyshev-based methods, the DCT-based method combined with the modified GCV criterion can reconstruct the force time history and identify the peak of impact force with high accuracy. Additionally, the identification of force location using the DCT-based method is also discussed.

Through combining P-type iterative learning (IL) control, model-free adaptive (MFA)control and sliding mode (SM) control, a robust model-free adaptive iterative learning (MFA-IL)control approach is presented for the active vibration control of piezoelectric smart structures.Considering the uncertainty of the interaction among actuators in the learning control process,MFA control is adopted to adaptively adjust the learning gain of the P-type IL control in order toimprove the convergence speed of feedback gain. In order to enhance the robustness of the systemand achieve fast response for error tracking, the SM control is integrated with the MFA control todesign the appropriate learning gain. Real-time feedback gains which are extracted fromcontrollers construct the basic probability functions (BPFs). The evidence theory is adopted to thedesign and experimental investigations on a piezoelectric smart cantilever plate are performed tovalidate the proposed control algorithm. The results demonstrate that the robust MFA-IL controlpresents a faster learning speed, higher robustness and better control performance in vibrationsuppression when compared with the P-type IL control.

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